Persistent homology provides information about the number of connected components, holes, trapped volumes, and higher-dimensional features in data across scales [1]}. Topaz, Ziegelmeier, and collaborators [2]}, [3]}, [4]} have used persistent homology to understand the shape of complex-systems data, and topological techniques are being increasingly applied to in vivo and in silico biological systems [5]}; examples include [4]}, [7]}, [8]}, [9]}, [10]}. Simplicial complexes are at the core of persistent homology, and we focus on one type of simplicial complex, namely the Vietoris–Rips complex. To build a Vietoris–Rips complex, we start with point-cloud data, such as the positions of \(N\)  cells in a domain. We place a ball with diameter \(\varepsilon \)  around each point and connect two points with an edge whenever their respective \(\varepsilon /2\) -balls intersect, indicating that the two points are at most a distance \(\varepsilon \)  apart [2]}, [4]}. If three points are all at most \(\varepsilon \)  apart, we note this by filling in the triangle bound by the edges between pairs of points. Data points are 0-simplices, edges connecting two points are 1-simplices, and filled-in triangles are 2-simplices [2]}, [4]}. We can continue this process to generate \(k\) -simplicies for \(k \ge 0\) , but we focus on \(k=0\)  and \(k =1\)  here.
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